Download Love of variety

4.3 Differentiated products
Vertical differentiation: different qualities
Horizontal differentiation: equal qualities, but
consumers perceive relevant differences.
Models:
Love of variety: representative consumer prefers n
varieties to n goods of the same variety.
Ideal variety: each consumer has a preferred variety.
Gießen, 03.12.2009
4.3.1 Love of variety
i = 1,…,N product varieties, N endogenous.
Utility function u(c1,…,cN) which is strictly concave in
ci implies “love of variety”.
Partial Equilibrium
Example 1: u(c) = v(ci), v’ > 0 > v”.
The FOCs for maximizing utility obtained from the
differentiated goods under the usual budget
constraint are
v‘(ci) = pi, i = 1,…,N
(20)
Gießen, 03.12.2009
v‘(ci) = pi, i = 1,…,N
(20)
Totally differentiating (20) and neglecting changes of
 (for large N) yields
v“dci = dpi  dci/dpi = /v“ < 0.
Elasticity of demand for variety i:
i = – [dci/dpi][pi/ci] = – v‘/civ“ > 0.
Assumption: di/dci < 0.
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Production: labor only factor, labor input:
Li = F + axi  ACi = wLi/xi = wF/xi + wa
Equilibrium conditions under monopolistic
competition (Chamberlin 1933):
w = p/a
Free entry  zero-profit condition
p = AC = wF/xi + wa  p/w = F/Lc + a
(21)
Monopoly pricing  MR = MC:
p[1 – 1/] = wa  p/w = a[/(1 – )]
(22)
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Figure 4.11: PP-curve: equ.(22), ZZ-curve: equ.(21)
Gießen, 03.12.2009
Equations (21) and (22) determine p/w and c. The
number of firms (= varieties) is obtained from labor
market clearing, hence
L = Li = (F + axi) = N(F + ax) = N(F + aLc)
N = 1/[F/L + ac].
Free trade between two identical countries:
equivalent to a doubling of L: downward shift of ZZcurve in figure 4.11: c (= quantity of each variety
consumed per consumer) falls, but real wage (w/p)
goes up, the number of varieties is increased,
number of firms per country is reduced.
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Predictions of the model:
 Real wage is increased
 Total number of varieties is increased
 Individual consumption of each variety falls.
 Number of firms per country falls.
 Output per surviving firm is increased.
Presumption: least efficient firms drop out.
Two cost reducing effects:
1. Scale effect
2. Selection effect
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 Agrees with argument in favor of free trade made
before trade agreements between U.S.A. and
Canada: Domestic market too small for firms to
operate at minimum efficient scale – free trade
allows to expand exports, but not all firms in all
countries can expand simultaneously – some
firms will drop out, the remaining ones will
produce more at lower average costs.
 However, not supported by empirical evidence
from U.S.A. and Canada free trade agreements:
No scale effects, only selection effects
Gießen, 03.12.2009
Example 2
CES-utility function (Dixit & Stiglitz 1977):
 n 1 
u c     ci 
 i 1

 1

Elasticity of substitution =  > 1, which is also equal
to  for large N. Let := /(  1). The
representative consumer solves
1
 n  
 n

max u c     ci     pi ci  I 
 i 1 
 n 1

Gießen, 03.12.2009
Implying the FOCs
u 1 
 
   ci 
ci   i 1 
1 
n

ci 1  pi  0.
 = marginal utility of income = v/I, i.e. partial
derivative of indirect utility function w.r.t. income.
The indirect utility function of a CES-function equals

 1
 n
v p, I   I   pi
 i 1





1 


 1
 n
     pi
 i 1





1 

For n very large, effect of ci on  and of pi on 
negligible.
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Define the constant k as

k
 n 1 
  ci 
 i 1 


1 

Substituting this into the FOC and re-arranging yields
1
1 
i
ci  p   kp
and thus
 = 1/[1 – ] = .
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The markup of prices over MC is fixed, i.e.
p/wa = /( − 1).
(23)
The profit equals
 = px – w(F + ax) = w[ax/( − 1) – F], and because of
the zero profit condition output per firm equals
x = ( − 1)F/a
(24)
The number of firms equals
N = L/(F + ax).
Thus, with free trade there is neither a scale effect
nor a selection effect, but the number of varieties
consumed is increased.
Gießen, 03.12.2009
General equilibrium
Assume there is also one homogeneous product,
denoted as y, and total utility equals
U(y,c) = y1-u(c).
 consumers devote a fraction  of their income to
buy differentiated products. Take the homogenous
product as numeraire good, i.e. py = 1.
Furthermore, factors of production are capital and
labor. All goods have constant MCs, but the
differentiated goods also have fixed costs F (e.g.
F = rKx).
Gießen, 03.12.2009
Integrated equilibrium with free entry:
1 = cy(w,r) = waLY + raKY
(25)
p = c(w,r,x) = waLX + raKX + rKX/x
(26)
p = [/( - 1)][waLX + raKX ]
(27)
aLYy + aLXNx = L
(28)
aKYy + aLXNx + NKX = K
(29)
 = pNx/[y + pNx]
(30)
Equations (25) and (26)are the zero profit conditions,
(27) is implied by profit maximization, (28) and (29)
are factor market clearing conditions, and (30) is
market clearing for the differentiated goods.
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Figure 4.12: FPE-set for differentiated goods (x) and
one homogenous good (y).
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• Free entry: FPE-set analogous to perfect
competition; inter-industrial trade AND intraindustrial trade.
• Embodied factor services flows according to
Heckscher-Ohlin model (factor abundance theory.
• Fixed number of firms: FPE-set analogous to
Cournot-oligopoly; country with larger number of
monopolistic firms may import labor and capital
services (paid for out of oligopoly rents).
Gießen, 03.12.2009
The gravity equation
Bilateral trade is directly proportional to the
product of the countries‘ GDPs.
Free trade equilibrium, all countries have identical
prices, identical production functions and identical
homothetic utility functions. C countries, N
products, all prices normalized to equal one 
GDP = Yi = k yik , world GDP = Yw = Yi.
sj = country j‘s share of world GDP = share of
world expenditure.
Gießen, 03.12.2009
Assuming that all countries are producing different
varieties export of country i to country j of product
k equals
Xkij = sjyik.
Summing over all products k yields
Xij = Xkij = sjyik = sjYj = YjYi/Yw = sjsiYw = Xji.
Summing the first and last term yields
Xij + Xji = [2/Yw]YiYj.
Explanation why trade grows faster than GDP.
Gießen, 03.12.2009